The value of `(1)/(sqrt(3.25)+sqrt(2.25))+(1)/(sqrt(4.25)+sqrt(3.25))+(1)/(sqrt(5.25)+sqrt(4.25))+(1)/(sqrt(6.25)+sqrt(5.25))` is

Options

A. `1.00`
B. `1.25`
C. `1.50`
D. `2.25`

Answer

To solve the expression

\[ \frac{1}{\sqrt{3.25} + \sqrt{2.25}} + \frac{1}{\sqrt{4.25} + \sqrt{3.25}} + \frac{1}{\sqrt{5.25} + \sqrt{4.25}} + \frac{1}{\sqrt{6.25} + \sqrt{5.25}}, \]

we will simplify each term step by step.

### Step 1: Simplify the first term

We start with the first term:

\[ \frac{1}{\sqrt{3.25} + \sqrt{2.25}}. \]

To simplify this, we can multiply the numerator and denominator by the conjugate of the denominator:

\[ \frac{1}{\sqrt{3.25} + \sqrt{2.25}} \cdot \frac{\sqrt{3.25} - \sqrt{2.25}}{\sqrt{3.25} - \sqrt{2.25}} = \frac{\sqrt{3.25} - \sqrt{2.25}}{(\sqrt{3.25})^2 - (\sqrt{2.25})^2}. \]

Calculating the squares:

\[ (\sqrt{3.25})^2 = 3.25 \quad \text{and} \quad (\sqrt{2.25})^2 = 2.25. \]

Thus,

\[ 3.25 - 2.25 = 1. \]

So, we have:

\[ \frac{\sqrt{3.25} - \sqrt{2.25}}{1} = \sqrt{3.25} - \sqrt{2.25}. \]

### Step 2: Simplify the second term

Now, we simplify the second term:

\[ \frac{1}{\sqrt{4.25} + \sqrt{3.25}}. \]

Using the same method:

\[ \frac{1}{\sqrt{4.25} + \sqrt{3.25}} \cdot \frac{\sqrt{4.25} - \sqrt{3.25}}{\sqrt{4.25} - \sqrt{3.25}} = \frac{\sqrt{4.25} - \sqrt{3.25}}{(\sqrt{4.25})^2 - (\sqrt{3.25})^2}. \]

Calculating the squares:

\[ (\sqrt{4.25})^2 = 4.25 \quad \text{and} \quad (\sqrt{3.25})^2 = 3.25. \]

Thus,

\[ 4.25 - 3.25 = 1. \]

So, we have:

\[ \frac{\sqrt{4.25} - \sqrt{3.25}}{1} = \sqrt{4.25} - \sqrt{3.25}. \]

### Step 3: Simplify the third term

Next, we simplify the third term:

\[ \frac{1}{\sqrt{5.25} + \sqrt{4.25}}. \]

Using the same method:

\[ \frac{1}{\sqrt{5.25} + \sqrt{4.25}} \cdot \frac{\sqrt{5.25} - \sqrt{4.25}}{\sqrt{5.25} - \sqrt{4.25}} = \frac{\sqrt{5.25} - \sqrt{4.25}}{(\sqrt{5.25})^2 - (\sqrt{4.25})^2}. \]

Calculating the squares:

\[ (\sqrt{5.25})^2 = 5.25 \quad \text{and} \quad (\sqrt{4.25})^2 = 4.25. \]

Thus,

\[ 5.25 - 4.25 = 1. \]

So, we have:

\[ \frac{\sqrt{5.25} - \sqrt{4.25}}{1} = \sqrt{5.25} - \sqrt{4.25}. \]

### Step 4: Simplify the fourth term

Finally, we simplify the fourth term:

\[ \frac{1}{\sqrt{6.25} + \sqrt{5.25}}. \]

Using the same method:

\[ \frac{1}{\sqrt{6.25} + \sqrt{5.25}} \cdot \frac{\sqrt{6.25} - \sqrt{5.25}}{\sqrt{6.25} - \sqrt{5.25}} = \frac{\sqrt{6.25} - \sqrt{5.25}}{(\sqrt{6.25})^2 - (\sqrt{5.25})^2}. \]

Calculating the squares:

\[ (\sqrt{6.25})^2 = 6.25 \quad \text{and} \quad (\sqrt{5.25})^2 = 5.25. \]

Thus,

\[ 6.25 - 5.25 = 1. \]

So, we have:

\[ \frac{\sqrt{6.25} - \sqrt{5.25}}{1} = \sqrt{6.25} - \sqrt{5.25}. \]

### Step 5: Combine all terms

Now we can combine all the simplified terms:

\[ (\sqrt{3.25} - \sqrt{2.25}) + (\sqrt{4.25} - \sqrt{3.25}) + (\sqrt{5.25} - \sqrt{4.25}) + (\sqrt{6.25} - \sqrt{5.25}). \]

Notice that this is a telescoping series. Most terms cancel out:

\[ -\sqrt{2.25} + \sqrt{6.25}. \]

Calculating the remaining terms:

\[ \sqrt{6.25} = 2.5 \quad \text{and} \quad \sqrt{2.25} = 1.5. \]

Thus,

\[ 2.5 - 1.5 = 1. \]

### Final Answer

The value of the expression is

\[ \boxed{1}. \]